ln a 1 ¼
ΔH m
R
1
T
0
f
À
1
T prep
!
ð6Þ
where ΔH m and T f
0 are the molar enthalpy of fusion and the normal freezing point
of pure solvent, respectively, and R is the gas constant.
2 The Flory–Rehner theory
including the ideal Donnan equilibria gives the following relation between the
activity of a swelling agent (solvent) in the gel and the gel parameters [77]:
lna 1 ¼ ln 1 À ν 2
ð
Þ þ ν 2 þ χν
2
2 þ ν e V 1 ν 2
1=3
ν
0
2
À Á 2=3 À 0:5 ν 2
h
i
À f ν 2 ð7Þ
where ν 2 and ν
0
2 are the volume fractions of crosslinked polymer in the equilibrium
swollen gel and after gel preparation, respectively, χ is the polymer–solvent
interaction parameter, ν e is the effective crosslink density of the network, V 1 is
the molar volume of solvent, and f is the effective charge density, i.e., the fraction of
charged units in the network chains that are effective in gel swelling. Since ν 2 ¼ ν
0
2
after cryogel preparation, substituting (7) into (6) yields [36]:
1
T prep
¼
1
T
0
f
À
R
ΔH m
ln 1 À ν
0
2
À
Á þ ν
0
2 þ χ ν
0
2
À Á 2 þ 0:5ν e V 1 ν
0
2 À f ν
0
2
h
i
ð8Þ
Using (8), one may estimate the polymer concentration ν
0
2 in the unfrozen gel
phase at a given temperature T prep (T prep < T f
0 ). To apply (8) to the cryogelation
systems, we assume that the monomers completely accumulate in the unfrozen
reaction phase before the onset of gelation. We also assume that, after cryogelation,
the conversion of monomer to crosslinked polymer is complete. Thus, the monomer
concentration C in the unfrozen zones (in % w/v) can be estimated from ν
0
2 as:
C ¼ ν
0
2 d 2
À
Á
10
2
ð9Þ
The volume fraction of frozen solvent in the reaction system, P, can also be
estimated as:
P ¼ 1 À
C o
C
ð10Þ
where C o is the nominal monomer concentration. Since the frozen solvent acts as a
template in the cryogels, P also corresponds to their porosities (volume of pores/
2 Assuming ideally dilute solutions, (6) leads to the well-known equation T prep À T
0
f ¼ À K f m 2 ,
where K f is the freezing-point-depression constant, and m 2 is the solute molality. However, this
equation cannot be used for the cryogelation system due to the high concentration of solutes in the
unfrozen domains.
120
O. Okay and V.I. Lozinsky
ΔH m
R
1
T
0
f
À
1
T prep
!
ð6Þ
where ΔH m and T f
0 are the molar enthalpy of fusion and the normal freezing point
of pure solvent, respectively, and R is the gas constant.
2 The Flory–Rehner theory
including the ideal Donnan equilibria gives the following relation between the
activity of a swelling agent (solvent) in the gel and the gel parameters [77]:
lna 1 ¼ ln 1 À ν 2
ð
Þ þ ν 2 þ χν
2
2 þ ν e V 1 ν 2
1=3
ν
0
2
À Á 2=3 À 0:5 ν 2
h
i
À f ν 2 ð7Þ
where ν 2 and ν
0
2 are the volume fractions of crosslinked polymer in the equilibrium
swollen gel and after gel preparation, respectively, χ is the polymer–solvent
interaction parameter, ν e is the effective crosslink density of the network, V 1 is
the molar volume of solvent, and f is the effective charge density, i.e., the fraction of
charged units in the network chains that are effective in gel swelling. Since ν 2 ¼ ν
0
2
after cryogel preparation, substituting (7) into (6) yields [36]:
1
T prep
¼
1
T
0
f
À
R
ΔH m
ln 1 À ν
0
2
À
Á þ ν
0
2 þ χ ν
0
2
À Á 2 þ 0:5ν e V 1 ν
0
2 À f ν
0
2
h
i
ð8Þ
Using (8), one may estimate the polymer concentration ν
0
2 in the unfrozen gel
phase at a given temperature T prep (T prep < T f
0 ). To apply (8) to the cryogelation
systems, we assume that the monomers completely accumulate in the unfrozen
reaction phase before the onset of gelation. We also assume that, after cryogelation,
the conversion of monomer to crosslinked polymer is complete. Thus, the monomer
concentration C in the unfrozen zones (in % w/v) can be estimated from ν
0
2 as:
C ¼ ν
0
2 d 2
À
Á
10
2
ð9Þ
The volume fraction of frozen solvent in the reaction system, P, can also be
estimated as:
P ¼ 1 À
C o
C
ð10Þ
where C o is the nominal monomer concentration. Since the frozen solvent acts as a
template in the cryogels, P also corresponds to their porosities (volume of pores/
2 Assuming ideally dilute solutions, (6) leads to the well-known equation T prep À T
0
f ¼ À K f m 2 ,
where K f is the freezing-point-depression constant, and m 2 is the solute molality. However, this
equation cannot be used for the cryogelation system due to the high concentration of solutes in the
unfrozen domains.
120
O. Okay and V.I. Lozinsky